Modal and temporal logic N. Bezhanishvili I. Hodkinson C. Kupke Imperial College London 1 / 83
Overview Part II 1 Soundness and completeness. Canonical models. 3 lectures. 2 Finite model property. Filtrations. 2 lectures. 3 Decidability. 2 lecture. 4 Modal µ -calculus. 2 lectures. 2 / 83
Syntactic approach Let us fix a class C of frames. We need to have, whenever possible, an effective criterion (algorithm) deducing whether a formula A is valid in C . If C is infinite then going through all the frames might take infinite time. Even if C is finite, but contains an infinite frame, the procedure might still take infinite time. In order to overcome this difficulty we will develop a syntactic (axiomatic) approach to modal logic. The idea of this approach is to find a small (possibly finite) number of formulas (axioms of our logic) and set some rules of inference which enable us to derive other formulas (theorems of our logic). 3 / 83
Idea The axioms are ‘given truths’. They should be valid (perhaps over some given class of frames). The rules allow us to derive new truths from old. If the formulas above the line in a rule are already derived, the rule allows us to derive the formula underneath the line. Rules should be chosen so as not to lead us from truth into falsehood. Formulas derived from valid formulas should also be valid. Here, we mean ‘valid in Kripke semantics’. Axiomatic approach is simple, powerful, flexible, and in common use. 4 / 83
Logics of classes of frames In Part I we defined when a formula A is valid in a class C of frames. A formula A is valid in a class C of frames if it is valid in every frame in C . That is, A is true in every element of every frame in C under every assignment. Given a class C of frames we can consider the set of all formulas valid in C . This set is called the logic of C and will be denoted by Log ( C ). In formal terms Log ( C ) = { A : A is a formula and ∀F ∈ C , A is valid in F} . 5 / 83
Let C be the class of all frames and let C ref be the class of all reflexive frames. That is, such frames F = ( W, R ) that for each x ∈ W we have R ( x, x ). What can we say about Log ( C ) and Log ( C ref )? Is any of these two sets included in the other? Proposition 1 Let C 1 and C 2 be classes of frames. If C 1 ⊆ C 2 , then Log ( C 2 ) ⊆ Log ( C 1 ) . Proof. Let A ∈ Log ( C 2 ). Then A is valid in every frame in C 2 . But since every frame in C 1 is in C 2 , the formula A is also valid in every frame in C 1 . This means that A ∈ Log ( C 1 ). 6 / 83
From the above proposition we deduce that Log ( C ) ⊆ Log ( C ref ). To show that the inverse inclusion does not hold we note that the formula ✷ A → A is valid in C ref , but is not valid in C . Therefore, the formula ✷ A → A belongs to Log ( C ref ), but does not belong to Log ( C ). Thus, these two logics are different. Exercise 2 Show that the logics of the classes of reflexive, transitive, serial and symmetric frames, respectively, are all different. If a class C consists of a single frame F , we denote the logic of C by Log ( F ) instead of Log ( {F} ). 7 / 83
Exercise 3 1 Show that if a frame F is a p -morphic image of a frame G , then Log ( G ) ⊆ Log ( F ) . 2 Show that Log ( N ) is contained in the logic of a single reflexive point. 3 Recall that a frame ( W, R ) is serial if for each s ∈ W there exists t ∈ W such that R ( s, t ) . Show that the logic of any class C of serial frames is contained in the logic of a single reflexive point. 8 / 83
Note that two different frame classes can have the same logic. For example, let C and C irref be the classes of all frames and all irreflexive frames, respectively. As we know from Part I (Lemmas 32 and 33) every frame is a p -morphic image of an irreflexive frame. Therefore, a formula is valid in the class of all frames iff it is valid in the class of all irreflexive frames. This means that Log ( C ) = Log ( C irref ). (This is just a different way of formulating Theorem 34 of Part I.) 9 / 83
Maximal logics There exists a characterization of the ‘maximal’ logics. Theorem 4 Let C be a non-empty class of frames. Then Log ( C ) is contained in the logic of a single reflexive point or Log ( C ) is contained in the logic of a single irreflexive point. Proof. Exercise. Use (3) of Exercise 3. This theorem is known as the Makinson theorem . 10 / 83
David Makinson (LSE) 11 / 83
Log ( ∅ ) Log ( F ) Log ( F ′ ) Log ( C ) Figure: The lattice of logics C is the class of all frames. F is one reflexive point, F ′ is one irreflexive point. 12 / 83
We have seen in Part I that for each A, B the formula ✷ ( A → B ) → ( ✷ A → ✷ B ) is valid in every Kripke frame. Exercise 5 1 Recall the proof of this fact. 2 Prove that if formulas A and A → B are valid in a frame F , then the formula B is also valid in F . 3 Prove that if a formula A is valid in a frame F , then the formula ✷ A is also valid in F . Now we will give a formal syntactic definition of (normal) modal logics. 13 / 83
By a propositional tautology in the modal language we will mean any ‘instance’ of valid propositional formulas (tautologies in the propositional language). E.g., p → ( q → p ) is a tautology in the propositional language. Therefore, A → ( B → A ) is a propositional tautology in the modal language (for each modal formulas A, B ). E.g., p ∨ ¬ p is a tautology in the propositional language. So, A ∨ ¬ A is a propositional tautology in the modal language (for each modal formula A ). Exercise 6 Are ✷ p ∨ ¬ ✷ p and ✷ p ∨ ✷ ¬ p propositional tautologies in the modal language? 14 / 83
Normal modal logics Definition 7 A normal modal logic (if it is clear from the context we will drop the worlds ‘normal’ and ‘modal’) L is a set of formulas that contains all propositional tautologies, the so called K -axioms: ✷ ( A → B ) → ( ✷ A → ✷ B ) and is closed under the rules of • modus ponens (MP): A, A → B B A • necessitation (N): ✷ A 15 / 83
This means that L is a set of formulas such that for each A, B the formula ✷ ( A → B ) → ( ✷ A → ✷ B ) ∈ L and (1) if A ∈ L and A → B ∈ L , then B ∈ L (2) if A ∈ L , then ✷ A ∈ L . Let L be a modal logic. Instead of A ∈ L we often write ⊢ L A and will read ‘ A is a theorem of L ’. The collection of all formulas is called the inconsistent logic . 16 / 83
Proposition 8 For each class C of frames, Log ( C ) is a normal modal logic. Proof. Exercise. The inconsistent logic is equal to Log ( ∅ ). Let K be the set of all formulas that we can generate starting from the propositional tautologies and K -axioms by applying (MP) and (N). We can show that K be the smallest normal modal logic (exercise). 17 / 83
This means that A ∈ K ( ⊢ K A ) iff if there are formulas A 1 , . . . A n such that • A n = A • each A i (1 ≤ i ≤ n ) is either – a propositional tautology – a K -axiom – or is obtained from some of A 1 , . . . , A i − 1 by (MP) or (N). 18 / 83
Example Let us prove ⊢ K ✷ ( A ∧ B ) → ✷ A . ⊢ K A ∧ B → A , (Propositional tautology) ⊢ K ✷ ( A ∧ B → A ), (N) ⊢ K ✷ ( A ∧ B → A ) → ( ✷ ( A ∧ B ) → ✷ A ), ( K -axiom) ⊢ K ✷ ( A ∧ B ) → ✷ A (MP). Exercise 9 1 Prove that ⊢ K A → B implies ⊢ K ✷ A → ✷ B . 2 Prove that ⊢ K A → B implies ⊢ K ✸ A → ✸ B . 3 Prove that ⊢ K ✷ ( A ∧ B ) ↔ ✷ A ∧ ✷ B . 4 Prove that ⊢ K ✸ ( A ∨ B ) ↔ ✸ A ∨ ✸ B . 19 / 83
We will be using the following simple proposition throughout. Proposition 10 1 If a modal formula A → B is a propositional tautology, then ⊢ K A implies ⊢ K B . 2 ⊢ K A and ⊢ K B imply ⊢ K A ∧ B . 3 If ⊢ K A → B and ⊢ K B → C , then ⊢ K A → C . 4 ⊢ K A → B and ⊢ K C → D , then ⊢ K ( A ∧ C ) → ( B ∧ D ) . Proof. We only prove (2) and (3). Proofs for (1) and (4) are similar. ⊢ K A → ( B → ( A ∧ B )), (Propositional tautology) ⊢ K A ∧ B , from ⊢ K A and ⊢ K B by applying (MP) twice. ⊢ K [( A → B ) ∧ ( B → C )] → ( A → C ), (Propositional tautology) ⊢ K A → C , by ⊢ K A → B , ⊢ K B → C , (2) and (MP). 20 / 83
Recall that C ref is the class of all reflexive frames . We want to find an axiomatic system that as we will see later characterizes Log ( C ref ). Let KT be the set of formulas that we can generate starting form K -axioms, the reflexivity axiom ✷ A → A and applying the rules of (MP) and (N). This we denote by KT = K + ( ✷ A → A ). Then KT is the smallest normal modal logic is the smallest extension of K that contains the reflexivity axiom (exercise). In the same way we can define the logics K4 = K + ( ✷ A → ✷✷ A ), S4 = K + ( ✷ A → A, ✷ A → ✷✷ A ) (‘ KT4 ’) If you also add A → ✷✸ A , you get S5 . See Chagrov and Zakharyaschev p. 116 or Blackburn et al. p. 193 for a longer list. 21 / 83
Definition 11 We say that a logic L is sound with respect to a frame class C if L ⊆ Log ( C ). That is, if every formula in L is valid in every frame in C . We say that a logic L is complete with respect to a frame class C if Log ( C ) ⊆ L . That is, if every formula valid in every frame in C belongs to L . Next we will discuss the canonical model construction . This construction is used for proving completeness of various normal modal logics and is certainly one of the most well-known and well-applied methods of modal logic. 22 / 83
Recommend
More recommend
